Problem: Let $\Omega$ a smooth, bounded open set of $\mathbb{R}^{N}$. Let a final time $T>0$, an initial data $u_{0} \in L^{2}(\Omega)$ and a source term $f \in L^{2}((0, T))$. Let $u$ the solution of $$ \left\{\begin{array}{l} \partial_{t} u-\Delta_{x} u=f, \text { a.e. in } \Omega \times(0, T), \\ u=0, \text { a.e. in } \delta \Omega \times(0, T), \\ u(0, x)=u_{0}(x), \text { a.e. in } \Omega . \end{array}\right. $$
Prove the following inequality, for $t \in[0, T]$ : $$ \begin{aligned} &\frac{1}{2} \int_{\Omega} u(t, x)^{2} d x+\int_{0}^{t} \int_{\Omega}\left|\partial_{x} u(s, x)\right|^{2} d x d s \\ &\quad \leq \frac{e^{t}}{2}\left(\int_{\Omega} u_{0}(x)^{2} d x+\int_{0}^{T} \int_{\Omega} f(s, x) u(s, x) d x d s\right) . \end{aligned} $$
My attempt: For $u$ is smooth I have shown that we have the following energy estimate: $$ \begin{aligned} &\frac{1}{2} \int_{\Omega} u(t, x)^{2} d x+\int_{0}^{t} \int_{\Omega}\left|\partial_{x} u(s, x)\right|^{2} d x d s \\ &\quad=\frac{1}{2} \int_{\Omega} u_{0}(x)^{2} d x+\int_{0}^{t} \int_{\Omega} f(s, x) u(s, x) d x d s\ (1) \end{aligned} $$ By applying the Gronwall Lemma I have also proved that for a function $z \in C^{0}\left([0, T], \mathbb{R}_{+}\right)$satisfying $$ z(t) \leq a+b \int_{0}^{t} z(s) d s, \quad \forall t \in[0, T], $$ where $a, b \geq 0$, we have $$ \forall t \in[0, T], \quad z(t) \leq a e^{b t}.\ (2) $$ Now, I intend to combine (1) and (2) with z is LHS of (1), $a= \frac{1}{2}\left(\int_{\Omega} u_{0}(x)^{2} d x+\int_{0}^{T} \int_{\Omega} f(s, x) u(s, x) d x d s\right)$ and $b=1$ to get what we need. However, it seems like we still have missed something to attain that. Anyone can show me what should I do to have the estimation above.